Article contents
Non-vanishing of class group L-functions for number fields with a small regulator
Published online by Cambridge University Press: 17 December 2020
Abstract
Let $E/\mathbb {Q}$ be a number field of degree
$n$. We show that if
$\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke
$L$-function does not vanish at the central point is
$\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in
$\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$ associated to the maximal order of
$E$, and the escape of mass of the torus orbit associated to the trivial ideal class.
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- Research Article
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- Copyright
- © The Author(s) 2020
References
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